TY - JOUR

T1 - Distortion of the hyperbolic robin capacity under a conformal mapping and extremal configurations

AU - Dittmar, B.

AU - Solynin, A. Y.

N1 - Funding Information:
The second author was supported by the Russian Foundation for Basic Research, grant 97-01-00259.

PY - 2002

Y1 - 2002

N2 - This paper is connected with recent results of Duren and Pfaltzgraff (J. Anal. Math., 78, 205-218 (1999)). We consider the problem on the distortion of the hyperbolic Robin capacity δ h (A, Ω) of the boundary set A ⊂ ∂Ω under a conformal mapping of a domain Ω ⊂ U into the unit disk U. It is shown that, for sets consisting of a finite number of boundary arcs or complete boundary components, the inequality cap hf(A) ≥ δh (A,Ω) (*) is sharp in the class of conformal mappings f: Ω → U such that f(∂U) = ∂U. Here cap h f(A) is the hyperbolic capacity of a compact set f(A) ⊂ U. We give some examples demonstrating properties of functions which realize the case of equality in relation (*).

AB - This paper is connected with recent results of Duren and Pfaltzgraff (J. Anal. Math., 78, 205-218 (1999)). We consider the problem on the distortion of the hyperbolic Robin capacity δ h (A, Ω) of the boundary set A ⊂ ∂Ω under a conformal mapping of a domain Ω ⊂ U into the unit disk U. It is shown that, for sets consisting of a finite number of boundary arcs or complete boundary components, the inequality cap hf(A) ≥ δh (A,Ω) (*) is sharp in the class of conformal mappings f: Ω → U such that f(∂U) = ∂U. Here cap h f(A) is the hyperbolic capacity of a compact set f(A) ⊂ U. We give some examples demonstrating properties of functions which realize the case of equality in relation (*).

UR - http://www.scopus.com/inward/record.url?scp=52649115462&partnerID=8YFLogxK

U2 - 10.1023/A:1015416110467

DO - 10.1023/A:1015416110467

M3 - Article

AN - SCOPUS:52649115462

VL - 110

SP - 3058

EP - 3069

JO - Journal of Mathematical Sciences

JF - Journal of Mathematical Sciences

SN - 1072-3374

IS - 6

ER -